Stability and the conservation of mass in drainage basin evolution

Stability and the conservation of mass in drainage basin evolution
Smith, Terence R.; Bretherton, Francis P.
1972-12-01 00:00:00
Drainage basin evolution is modeled as the time development of an initial surface subject to conservation of sediment and water and a transport law qs = F(S, q) connecting the sediment flux qs with the local slope S and the discharge of surface water q. Two models are presented. The first is appropriate to a smooth surface on which no discrete channels have formed, and the second is appropriate to a family of V‐shaped valleys, each containing a separate stream of negligible width. For the first model, some solutions are presented that describe the evolution of a long ridge for which the profile is independent of one spatial coordinate. The stability of such surfaces is then discussed. It is shown that, if F/q < ∂F/∂q, disturbances of small amplitude and small lateral scale will grow rapidly and presumably will lead to the formation of closely spaced channels directed down the slope, whereas, if F/q ≥ ∂F/∂q, such channels will tend to disappear. For a surface eroding without change of shape, convex portions are stable, and concave segments are unstable. The second model is less well developed, since conservation principles alone are insufficient to determine its evolution without some additional postulate describing the sideways migration of individual streams. It is shown how, if each stream moves so that the sediment fluxes entering from its two side slopes remain equal, a system of similar parallel valleys is unstable, and neighbors will tend to coalesce on a time scale comparable to that for erosion through a layer as thick as a valley is deep.
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Abstract

Drainage basin evolution is modeled as the time development of an initial surface subject to conservation of sediment and water and a transport law qs = F(S, q) connecting the sediment flux qs with the local slope S and the discharge of surface water q. Two models are presented. The first is appropriate to a smooth surface on which no discrete channels have formed, and the second is appropriate to a family of V‐shaped valleys, each containing a separate stream of negligible width. For the first model, some solutions are presented that describe the evolution of a long ridge for which the profile is independent of one spatial coordinate. The stability of such surfaces is then discussed. It is shown that, if F/q < ∂F/∂q, disturbances of small amplitude and small lateral scale will grow rapidly and presumably will lead to the formation of closely spaced channels directed down the slope, whereas, if F/q ≥ ∂F/∂q, such channels will tend to disappear. For a surface eroding without change of shape, convex portions are stable, and concave segments are unstable. The second model is less well developed, since conservation principles alone are insufficient to determine its evolution without some additional postulate describing the sideways migration of individual streams. It is shown how, if each stream moves so that the sediment fluxes entering from its two side slopes remain equal, a system of similar parallel valleys is unstable, and neighbors will tend to coalesce on a time scale comparable to that for erosion through a layer as thick as a valley is deep.